*Preprint*

**Inserted:** 16 may 2024

**Year:** 2024

**Abstract:**

We show a sharp and rigid spectral generalization of the classical Bishop--Gromov volume comparison theorem: if a closed Riemannian manifold $(M,g)$ of dimension $n\geqslant 3$ satisfies \[\lambda_1(-\gamma\Delta+\mathrm{Ric})\geqslant n-1\] for some $0\leqslant\gamma\leqslant\frac{n-1}{n-2}$, then $\operatorname{vol}(M)\leqslant\operatorname{vol}(\mathbb S^{n})$, and $\pi_1(M)$ is finite. Moreover, the bound on $\gamma$ is sharp for this result to hold. A generalization of the Bonnet--Myers theorem is also shown under the same spectral condition. The proofs involve the use of a new unequally weighted isoperimetric problem and unequally warped $\mu$-bubbles. As an application, in dimensions $3\leqslant n\leqslant 5$, we infer sharp results on the isoperimetric structure at infinity of complete manifolds with nonnegative Ricci curvature and uniformly positive biRicci curvature.